Measure (mathematics)

Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (seeDefinition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.[1] Indeed, their existence is a non-trivial consequence of the axiom of choice.

which implies (since the sum on the right thus converges to a finite value) that μ(∅)=0{\displaystyle \mu (\varnothing )=0}.

If only the second and third conditions of the definition of measure above are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.

The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets. If (X,ΣX){\displaystyle \left(X,\Sigma _{X}\right)} and (Y,ΣY){\displaystyle \left(Y,\Sigma _{Y}\right)} are two measurable spaces, then a function f:X→Y{\displaystyle f:X\to Y} is called measurable if for every Y-measurable set B∈ΣY{\displaystyle B\in \Sigma _{Y}}, the inverse image is X-measurable – i.e.: f(−1)(B)∈ΣX{\displaystyle f^{(-1)}(B)\in \Sigma _{X}}. The composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows.

A measure μ is continuous from above: If E1, E2, E3, ..., are measurable sets and for all n, En + 1 ⊂ En, then the intersection of the sets En is measurable; furthermore, if at least one of the En has finite measure, then

This property is false without the assumption that at least one of the En has finite measure. For instance, for each n ∈ N, let En = [n, ∞) ⊂ R, which all have infinite Lebesgue measure, but the intersection is empty.

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure 1μ(X)μ{\displaystyle {\frac {1}{\mu (X)}}\mu }. A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals[k, k+1] for all integersk; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

A measurable set X is called a null set if μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I{\displaystyle I} and any set of nonnegative ri,i∈I{\displaystyle r_{i},i\in I} define:

That is, we define the sum of the ri{\displaystyle r_{i}} to be the supremum of all the sums of finitely many of them.

A measure μ{\displaystyle \mu } on Σ{\displaystyle \Sigma } is κ{\displaystyle \kappa }-additive if for any λ<κ{\displaystyle \lambda <\kappa } and any family of disjoint sets Xα,α<λ{\displaystyle X_{\alpha },\alpha <\lambda } the following hold:

Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice.

A charge is a generalization in both directions: it is a finitely additive, signed measure.